lmomtexp: L-moments of the Truncated Exponential Distribution

Description

This function estimates the L-moments of the Truncated Exponential distribution. The parameter $\psi$ is the right truncation of the distribution and $\alpha$ is a scale parameter, letting $\beta = 1/\alpha$ to match nomenclature of Vogel and others (2008), the L-moments in terms of the parameters, letting $\eta = \mathrm{exp}(-\alpha\psi)$, are $$\lambda_1 = \frac{1}{\beta} - \frac{\psi\eta}{1-\eta} \mbox{,}$$ $$\lambda_2 = \frac{1}{1-\eta}\biggl[\frac{1+\eta}{2\beta} - \frac{\psi\eta}{1-\eta}\biggr] \mbox{,}$$ $$\lambda_3 = \frac{1}{(1-\eta)^2}\biggl[\frac{1+10\eta+\eta^2}{6\alpha} - \frac{\psi\eta(1+\eta)}{1-\eta}\biggr] \mbox{, and}$$ $$\lambda_4 = \frac{1}{(1-\eta)^3}\biggl[\frac{1+29\eta+29\eta^2+\eta^3}{12\alpha} - \frac{\psi\eta(1+3\eta+\eta^2)}{1-\eta}\biggr] \mbox{.}$$

The distribution is restricted to a narrow range of L-CV ($\tau_2 = \lambda_2/\lambda_1$). If $\tau_2 = 1/3$, the process represented is a stationary Poisson for which the probability density function is simply the uniform distribution and $f(x) = 1/\psi$. If $\tau_2 = 1/2$, then the distribution is represented as the usual exponential distribution with a location parameter of zero and a scale parameter $1/\beta$. Both of these limiting conditions are supported.

If the distribution shows to be Uniform ($\tau_2 = 1/3$), then $\lambda_1 = \psi/2$, $\lambda_2 = \psi/6$, $\tau_3 = 0$, and $\tau_4 = 0$. If the distribution shows to be Exponential ($\tau_2 = 1/2$), then $\lambda_1 = \alpha$, $\lambda_2 = \alpha/2$, $\tau_3 = 1/3$ and $\tau_4 = 1/6$.

Usage

lmomtexp(para)

Value

An R

list is returned.

lambdas: Vector of the L-moments. First element is $\lambda_1$, second element is $\lambda_2$, and so on.
ratios: Vector of the L-moment ratios. Second element is $\tau$, third element is $\tau_3$ and so on.
trim: Level of symmetrical trimming used in the computation, which is 0.
leftrim: Level of left-tail trimming used in the computation, which is NULL.
rightrim: Level of right-tail trimming used in the computation, which is NULL.
source: An attribute identifying the computational source of the L-moments: “lmomtexp”.

Arguments

para: The parameters of the distribution.

Author

W.H. Asquith

References

Vogel, R.M., Hosking, J.R.M., Elphick, C.S., Roberts, D.L., and Reed, J.M., 2008, Goodness of fit of probability distributions for sightings as species approach extinction: Bulletin of Mathematical Biology, DOI 10.1007/s11538-008-9377-3, 19 p.

Examples

Run this code

set.seed(1) # to get a suitable L-CV
X <- rexp(1000, rate=.001) + 100
Y <- X[X <= 2000]
lmr <- lmoms(Y)

print(lmr$lambdas)
print(lmomtexp(partexp(lmr))$lambdas)

print(lmr$ratios)
print(lmomtexp(partexp(lmr))$ratios)

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